Aircraft: SMALL AIRCRAFT; LOW WING; 1 ENG; FIXED GEAR
Phase of flight: Cruise
Anomaly: ATC Issue All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Published Material / Policy

The staffing has been really poor. We have taken a few hits because of people being out with coronavirus or close contact. I just wanna say on Day X that I worked a X AM to Y PM shift and that was the beginning of my week. It was tough to get caught up from the fatigue of that for a shift in the very first part of my week. Also; I have worked X weeks in a row with only a single day off.I took a handoff Aircraft X IFR from ZZZ [Center] (X SECTOR) at 4;000 destined ZZZ [Airport]. Airspace configuration was north based on ZZZ1 [Airport] international traffic. This required me to vector Aircraft X around the ZZZ1 [Airport] Final Box. I did so. Because of the staffing we were told or encouraged to not work VFR overflights or VFR practice approaches that day. I don't know if that word was past two adjacent facilities; But I was trying to eliminate as much of the VFR traffic as I could by terminating them whenever they checked on from other facilities or by denying services.ZZZ [Center] (Y SECTOR) handed off Aircraft Y; VFR at 4;500; but the pilot never checked in. I had to call ZZZ [Center] (Y SECTOR) and ask for communications. I had no intention of working Aircraft Y. I wanted to be able to communicate with the pilot to terminate Flight Following in order to concentrate on other tasks. I had the Flight Data controller remove Aircraft Y from the NAS.While I'm doing this; a couple of things happened at or about the same time. Aircraft Z checked on the frequency and was in the ZZZ1 [Airport] Final Box under the Class Bravo Airspace. Aircraft Z requested clearance into the Class Bravo airspace and to navigate VFR to a hospital that I hadn't heard of and was not mapped. I was trying to determine where exactly the hospital was. At or about that same time I had an aircraft (Aircraft A) check on my frequency (the wrong frequency); wanting clearance into the Class Charlie airspace at ZZZ2 [Airport] for closed traffic. Aircraft A should have made the request on ZZZ1 [TRACON] frequency XYZ.Y. I looked over to my left and the ZZZ1 [TRACON] Controller looked and sounded busy so I decided to accommodate Aircraft A and then point out Aircraft A to the ZZZ1 [TRACON] Controller. I issued all of the instructions for Aircraft A. I also accomplished all of the coordination for the Aircraft Z with the Final Controller. When I finally get Aircraft Y on frequency I told him I was too busy to accommodate VFR flight following; told him squawk VFR and radar service terminated. At some point I drop the track on the wrong aircraft. I drop the track on Aircraft X IFR to ZZZ [Airport].I'm shocked that I did not realize that I dropped the track on the wrong flight; but for whatever reason I did not. I'm pretty sure I was pretty tired by this point. My previous session in the day ran almost 2 hours; during which I took an absolute pounding. I am further shocked that at no time in the session did I realize and correct my mistake.Later in the session I took the hand off on Aircraft B from ZZZ1 [TRACON] Y Sector. Aircraft B was requesting 6;000. Departure 3 & 4 were being worked by one controller; who was probably the busiest controller in the room. I decided to give Aircraft B; 4;000 as a final altitude.Aircraft B was now IFR at 4;000 southwest bound and on a converging course with Aircraft X. I don't remember the destination for Aircraft B. Aircraft X is tracking pretty much direct ZZZ [Airport] I don't remember if I had given them direct or not. I noticed Aircraft B was converging with limited target at 4;000; which was Aircraft X. I called the traffic to Aircraft X. Seeing the conflict I decided to climb Aircraft B to 5;000. I don't recall if Aircraft B ever called the traffic in sight; but I know; after watching the replay that I never had more than 500 feet separation between the two flights.Still not realizing my mistake; Aircraft X calls ZZZ [Airport] in sight. I ask who he is and where he came from. Not one of my better moments. At some point; I began to remember Aircraft X and I start the track on the flight; make the appropriate entries in the scratch pad; coordinate with the ZZZ1 [TRACON] Y Sector Controller for landing ZZZ [Airport]; inbounded the flight to the Tower; and cleared Aircraft X for the visual approach to ZZZ [Airport].I know this recommendation will be laughed out of the office; however it would be terrific if the terminal world had a version of EDST [En-route Decision Support Tool] along with a monitor/terminal to view and amend flight data at your position. This technological implementation is frankly overdue and it is a genuine travesty that we don't have it. TRACON Controller reported numerous issues while working traffic and problems associated with the workload and staffing.